A point is called a branch-point if analytic continuation over a closed curve around it can produce a different value upon reaching the starting point. Take for example starting at the point and analytically continuing the function around the unit circle using the differential equation (just differentiate the function and let ). Upon integrating from zero to , . Therefore, there is a branch point in the unit circle.

A multivalued function is a complex function that for a given can assume several different values. Typical examples are 'nth root', logarithm, inverse circular functions and so on. A branch point of a multivalued function is a point in the complex plane from which depart two or more branches of a multivalued function. Let consider for example the multivalued function . The point is a branch point for it because, setting , is...

(1)

In (1) the sign '+' is for k even and the sign '-' for k odd and at different signs they correspond two different branches that have in common the point , that for this reason is called 'branch point'...